APSS Apollo Application Note on Photonic Crystal Fibers (PCFs)

Transcription

1 APSS Apollo Application Note on Photonic Crystal Fibers (PCFs) Design and simulation APN-APSS-PCF Apollo Inc Main Street West Hamilton, Ontario L8S 1B7 Canada Tel: (905) Fax: (905)

2 Disclaimer In no event should Apollo Inc., its employees, its contractors, or the authors of this documentation be liable to you for general, special, direct, indirect, incidental or consequential damages, losses, costs, charges, claims, demands, or claim for lost profits, fees, or expenses of any nature or kind. Document Revision: July 30, 2003 Copyright 2003 Apollo Inc. All right reserved. No part of this document may be reproduced, modified or redistributed in any form or by whatever means without prior written approval of Apollo Inc. Abstract Apollo Inc. Page 2 of 21 APN-APSS-PCF

3 This application note provides an overview and an example of how to design, simulate, and optimize photonic crystal fibers (PCFs) using the Waveguide Module of the Apollo Photonics Solution Suite (APSS). This application note: describes the operation principle and unique modal characteristics of PCFs presents the basic design process for PCF-based waveguides in the context of specific applications discusses key issues related to PCF-based waveguides, such as single mode operation, dispersion, mode effective area, confinement loss, and polarization dependence outlines design steps specific to the design of PCF-based waveguides, such as import projects, waveguide implementation, solver settings, and display of simulation results provides some examples to illustrate the applications of PCFs and simulation results, which can then be compared with published papers The APSS application consists of four different modules: Material, Waveguide, Device, and Circuit. Because each module specializes in different specific design tasks, APSS can handle almost any kind of device made from almost any kind of material. Keywords APSS, waveguide module, photonic crystal fibers (PCFs) waveguide, single mode operation, photonic band gap, dispersion, mode effective area, waveguide implementation, vector finite difference method. Apollo Inc. Page 3 of 21 APN-APSS-PCF

4 Table of Contents 1 INTRODUCTION THEORY OPERATION PRINCIPLE PERFORMANCE PARAMETERS BASIC DESIGN CONSIDERATIONS Analysis methods Scaling transformations Complicated structures DESIGN AND SIMULATION OVERALL DESIGN EFFECTIVE AREA RELATED PCFS DISPERSION RELATED PCFS (DSFS,DFFS, AND DCFS) SIMULATION AND OPTIMIZATION EXAMPLES MATERIAL DESIGN CREATION OF PCF IN A USER-DEFINED WAVEGUIDE SOLVER SETTINGS FOR THE WAVEGUIDE RUN AND DISPLAY CONCLUSION REFERENCES Apollo Inc. Page 4 of 21 APN-APSS-PCF

5 1 Introduction Photonic crystal fibers [1] (PCFs, also called the holley fibers, or HFs; and the microstructured fibers, or MOFs) constitute a new class of photonic crystal waveguides, formed by introducing a high-index defect (or a missing hole) in a regular triangular (or hexagonal) photonic crystal (PC) array [2] of air holes along the propagation direction of the waveguide. Unlike the conventional fibers, the effective cladding index of the PCFs is a strong function of wavelength, and the modes supported by the waveguides are generally more dispersive. For this reason, PCFs have some unique properties that cannot be achieved in conventional fibers, such as single-mode operation at a wide wavelength range, highly tunable dispersion, and highly controllable mode effective areas for linear and nonlinear applications. Because of its unique properties and ease of fabrication, PCFs can be made from a number of optical materials, such as silica and polymer; and they have many potential applications such as dispersion-related fibers [3], supercontinuum generation, and large mode effective area fibers. The optical properties of PCFs are easily controlled by filling the air holes with thermosensitive material. Based on the photonic band structure of the PC array and the appropriate boundary conditions, PCFs can be analyzed by using several theoretical models. These models range from simpler analytical approaches, such as effective index method and envelope approximation method, to time-intensive, more rigorous numerical approaches, such as the finite difference method (FDM) and finite-difference timedomain method (FDTD) [1]. The primary method used for this application note is the finite difference method. Note: For more information about available categories of waveguides and solver settings for the Waveguide Module, refer to APSS Waveguide Module user manual. For more information about basic design requirements, such as single mode condition and polarization-dependence, please refer to other related application notes for the APSS. Apollo Inc. Page 5 of 21 APN-APSS-PCF

6 2 Theory In this section, the operation principle and unique modal characteristics of PCFs are described. Some basic design considerations and performance parameters for PCFs are also provided. 2.1 Operation principle Figure 1 shows the transverse cross-section of a typical PCF that consists of a central high-index defect (or a missing hole) in a regular triangular (or hexagonal) array of air holes. Except the number N of rings of air holes, there are two main design parameters, namely, the air hole diameter d and the pitch Λ. Y X Figure 1 The cross section of a PCF consisting of a regular triangular air hole array with five rings (N=5) of air holes with two physical parameters: the air hole diameter d and the pitch Λ Like conventional step index fibers, the operation principle for PCFs is to guide light by the total internal reflection (TIR). Another similarity between step index fibers and PCFs is that the effective refractive index in the area around the core is lower than that of the core itself. From the band structure of the PC cladding, the effective index n FSM1 of the lowest band (also called the fundamental space filling mode, or SFM1), which is a strong Apollo Inc. Page 6 of 21 APN-APSS-PCF

7 function of wavelength, can be calculated. Therefore, for the guided modes, their corresponding effective indices n eff meet the following relation: n > n > n Eq. 1 c eff FSM1 where n c is the refractive index of the core of the waveguide (for example, pure silica) and n FSM1 is the cladding effective index of the PCF. In order to use the modal properties of conventional step index fibers, the normalized effective frequency V eff, less than for the single mode operation, is defined as: V eff = 2π a λ eff n 2 c n 2 FSM1 Eq. 2 where a eff is the equivalent core radius of PCFs and λ is the operating wavelength. It is worth noting: that the range of the equivalent core radius a eff could be selected from Λ/2 to Λ, where Λ is the pitch of PCFs because the cladding effective index has a strong wavelength dependence, the equivalent core radius a eff is another variable that depends on the wavelength and design parameters of PCFs Overall, however, the operation principle of PCFs is based on the TIR effect. For sake of simplicity, the regular circular PCF with one missing hole, as shown in Figure 1, is used as the main example in this application note. However, more complicated PCFs (for example, those that have interstitial holes, oval shapes for polarization-maintaining fibers, or more missing holes) can be designed following the same process. For more information about photonic band gap (PBG) fibers and the photonic crystal waveguides (PCWs), in which the effective index of the cladding is always higher than that of the core, and whose guidance is due to the PBG effect of the periodic material. 2.2 Performance parameters The parameters for the PCF include the propagation constant, loss, group delay, dispersion, effective area, and modal diameters, and these are all calculated in the APSS. Note: For more information about these parameters, please refer to Appendix IV of APSS user manual. Apollo Inc. Page 7 of 21 APN-APSS-PCF

8 2.3 Basic design considerations PCF FIBER When the operation principle of PCFs and the relevant performance parameters for the materials to be used (for example, silica or polymer) have been understood, and the design requirements (for example, single mode operation and dispersion), a designer can apply analytical and numerical solvers to design PCFs Analysis methods There are several methods that can be used to analyze the modal characteristics of PCFs. There are two classes of methods: analytical and semi-analytical methods, such as the effective index method (EIM), the envelope approximation method (EAM), and the modal expansion methods (for example, the plane wave expansion method, or PWE) numerical methods, such as the finite difference method (FDM), the finiteelement method (FEM), and the finite difference time domain method (FDTD). In order to understand and fabricate effective PCFs, both analytical and numerical methods should be used to predict accurate modal properties. For example, EIM treats the PCF as a waveguide in which a silica core (or defect) is surrounded by a uniform cladding with a lower effective index, which can be related to the band structure of the photonic crystal in absence of the core. In other words, the main task in EIM is to calculate the effective index of the PC cladding associated with the space-filling modes. It is simple and intuitive and the modal properties of the PCF can be solved by conventional waveguide theory as long as the effective cladding index is obtained. However, due to the inherent assumption that the photonic crystal structure is extended to infinity, the effects of the finite number of the rings and special shapes of the PCF cannot be accounted for. Fortunately, all of these issues are addressed by numerical methods. The APSS application uses the numerical method FDM, which is simple and powerful. If a designer wants to use EIM or EAM, the band structure can be calculated prior to using APSS, so that FDM can be used in APSS to calculate the final modal properties. Apollo Inc. Page 8 of 21 APN-APSS-PCF

9 2.3.2 Scaling transformations There is no fundamental length scale for the EM wave because of the nature of Maxwell s equations. For this reason, the scaling transformations of the mode field pattern and the effective index are obtained by scaling the position r and frequency ω by the same factor M [2]: n λ λ, M ) fixed Λ neff ( ) Eq. 3 M eff ( d/ = v v r λ E( r, λ, M ) fixed d/ Λ = E(, ) Eq. 4 M M v v r λ H ( r, λ, M ) fixed d/ Λ = H (, ) Eq. 5 M M where d/λ is the size-to-pitch ratio of PCFs, λ is the operating wavelength, and M is the pitch ratio (= Λ/Λ 0, for example,λ 0 = 2.3 µm). Once the scaling transformations of the mode field pattern and the effective index are given, the scaling transformations related to other modal parameters can be easily obtained [3]: D A L 1 λ λ, M ) fixed Λ Dg ( ) Eq. 6 M M g ( d/ = 2 λ ( λ, M ) fixed d/ Λ M Aeff ( ) Eq. 7 M eff = 1 λ λ, M ) fixed Λ Lc ( ) Eq. 8 M M c ( d/ = λ Γ( λ, M ) fixed d/ Λ = Γ( ) Eq. 9 M In similar way, scaling approximations of the modal properties of PCFs with the fixed pitch ratio Λ can be approximately calculated [3]: λ Dg ( λ, N) fixed Λ A( N) Dg ( ) Eq. 10 B(N) A eff 1 λ ( λ, N) fixed Λ Aeff ( ) Eq. 11 C( N) D(N) 1 λ Lc ( λ, N) fixed Λ Lc ( ) Eq. 12 E( N) F( N) Apollo Inc. Page 9 of 21 APN-APSS-PCF

10 λ Γ( λ, N) fixed d/ Λ Γ( ) Eq. 13 G( N) where N is the air hole diameter ratio (= d/d 0, for example, d 0 = 1.15 µm). For small-airhole PCFs (d/λ< 0.5), the scaling coefficients A, B, C, D, E, F, and G can be linearly approximate (for example, N). However, for large-air-hole PCFs, the scaling coefficient A, B, C, D, E, F, and G are approximately obtained through the nonlinear function of N [3]. By using scaling transformations, modal properties of PCFs with any d, Λ, and λ are easily calculated and these approximate scaling transformations can assist in achieving a PCF design that has the desired properties Complicated structures PCFs are generally easier to design for certain applications, including those that are dispersion-related or require a large mode effective area. However, in order to take advantage of the unique properties of PCFs, such as large nonlinear and birefringence effects, more complicated designs can be developed that might incorporate special shapes, different arrangements of shapes, multi-cored structures and interstitial holes. Fortunately, the design of more complicated PCFs can be done by following the same design procedure covered in this document, even though a very simple example is provided here. Note that the two fundamental modes of PCFs are degenerate due to its π/3 symmetry. 3 Design and simulation 3.1 Overall design This section introduces a general procedure for designing PCFs with desired modal properties. According to some related design experiences (for example, from dispersionrelated applications [3]), the following process should be used: (i) Target function with proper constraints given a set of desirable modal properties for specific wavelengths, define a target function with constraints. Apollo Inc. Page 10 of 21 APN-APSS-PCF

11 (ii) Preliminary optimization based on scaling transformations with varying air hole diameter d and pitch Λ, optimize the PCF by minimizing the target function through scaling transformations. (iii) Model refinement for the modal properties with the optimized PCF obtained in (ii), calculate modal properties of the PCF using rigorous vector solvers at the required wavelength range and repeat (ii). (iv) Verification of the final design with the optimized PCF obtained in (iii), calculate the modal properties of the PCF through the rigorous vector solvers with high accuracy setting at the required wavelength range to check if the optimized PCF design meets the required dispersion properties. If so, calculate other modal parameters and end the design. If not, change the target function and repeat (ii)- (iv). In general, the final PCF structure can be obtained after two or three refinements achieved through using the vector solvers. 3.2 Effective area related PCFs In review, by changing the design parameters, the mode effective area of PCFs can vary from 1.0 to µm 2. Figure 2 shows a typical mode effective area for the fixed pitch Λ. According to the corresponding requirements, the target functions can be easily obtained and final results are readily optimized. Apollo Inc. Page 11 of 21 APN-APSS-PCF

12 Mode effective area A eff (µm 2 ) d = 0.46 µm d = 0.69 µm 0.92 µm 1.15 µm 1.38 µm 2.3 µm Wavelength λ (µm) Figure 2 Mode effective area A eff as a function of wavelength for different PCFs with fixed Λ = 2.3 µm 3.3 Dispersion related PCFs (DSFs,DFFs, and DCFs) Because of their highly dispersive properties, PCFs can be used in dispersion-related applications such as the dispersion-shifted fibers (DSFs), dispersion-flattened fibers (DFFs), and dispersion-compensation fibers (DCFs). Figure 3 shows the typical dispersion curve of a PCF, which is reasonable for PCFs with d/λ > 0.2, with dispersion wavelengths λ D1, λ D2, and λ D3 for the required dispersion D F (that is, D( λ ) = D, i = 1,2,3 ) as a function of wavelength. According to the corresponding Di F dispersion requirements, the target functions are easily obtained and final results are readily optimized. Apollo Inc. Page 12 of 21 APN-APSS-PCF

13 Total dispersion D (ps/nm/km) λ D1 λ S1 λ D2 λ F D = D F λ S2 λ D3 Wavelength λ (µm) Figure 3 Total dispersion D of PCFs as a function of wavelength with some possible dispersion wavelengths λ D1, λ D2, λ D3, two wavelengths λ S1, λ S2 of zero third-order dispersion, and one wavelength λ F of zero fourth-order dispersion 3.4 Simulation and optimization Using APSS, an optical designer can create and simulate whole structures or half structures. The benefit of designing a half structure is that the APSS application can then take advantage of the symmetry of the waveguide, saving simulation time. However, to analyze bend-related properties, the whole structure must be analyzed. In the user-defined model, the designer can build the PCF using different shapes and functions. For a typical PCF like the one shown in Figure 1, only the Ellipse shape and Array function would be used. Table 1 shows the related data for the Ellipse shape and Array function of the half structure. Figure 4 shows the 11-by-5-circle PCF with two design parameters d (for example, =1.84 µm by default) and the pitch Λ (for example, =2.3 µm by default). Note: For more specific information about how to use the APSS Waveguide Module, refer to the APSS Waveguide Apollo Inc. Page 13 of 21 APN-APSS-PCF

14 Module in APSS User Manual and Getting Started. Table 1 The related data for the Ellipse shape and Array function of the half structure PCF size 7-by-3-circle 11-by-5-circle 15-by-7-circle Variable The pitch R, the air hole radius d, half X width: W Window size X 6S+2d 9S+2d 14S+2d Window size Y 3R 5R 7R First circle size d, d, W-3S, R/2 d, d, W-5S, R/2 d, d, W-7S, R/2 Array with distance between 7x8 (2 S, R) 5x6 (2 S, R) 3x4 (2 S, R) shapes Second circle size d, d, W-2S, R d, d, W-4S, R d, d, W-6S, R Array wjith distance between shapes 6x7 (2 S, R) 4x5 (2 S, R) 2x3 (2 S, R) Note: the middle parameter S=R/2tan(π/3.0) and here d is the radius of the air hole. Apollo Inc. Page 14 of 21 APN-APSS-PCF

15 Figure 4 The 11-by-5 circle PCF with two design parameters: the air hole size d (for example, =1.84 µm by default) and the pitch Λ (for example, =2.3 µm by default) After building the PCF, it can be simulated and scanned for any related variables. There are many choices for the device solver settings. For example, you can select a solver with the settings Real or Complex, Direct or Iterlative, without and with PML. Note: For more information about the solver options and waveguide analysis tools available in the Waveguide Module, (such as for calculating the overlap integral between different waveguides and the far-field/near-field calculations of the waveguide project), refer to the APSS Apollo Inc. Page 15 of 21 APN-APSS-PCF

16 User Manual, or APSS Getting Started manual. Finally the user can display the simulation results to view the waveguide performance parameters such as the modal field patterns, the effective index, the propagation constant, dispersion, mode effective area, modal diameter, and loss. The user can also export them in different formats such as ASCII text (*.txt), Microsoft Excel (*.xls), or as a bitmap (*.bmp) file. 4 EXAMPLES The APSS Waveguide Module, as outlined in the previous sections, is a convenient platform for designing and simulating PCFs with specific properties. Following are some examples of how the Waveguide Module can be used for similar projects that use the same silica material. 4.1 Material Design As mentioned before, before starting to design the PCF, the user should complete the material design. In APSS, there are two pre-defined material systems (InP and silica) in the Material Module, which cover most applications, as well as a user-defined editor to accommodate more complicated or new applications. In this application, in order to compare the simulation results with a published paper, pure silica material is used through the three-term Sellmeier formula [4]. The first step is to create a material project, M_Silica_3_0.5_6_111 with one material for wavelength range of µm, as shown in Figure 5. Apollo Inc. Page 16 of 21 APN-APSS-PCF

17 Figure 5 The silica material project for the PCF waveguide 4.2 Creation of PCF in a user-defined waveguide In this example, the design parameters from [1] were used build the PCF with two design parameters: the air hole size d (for example, =1.84 µm by default) and the pitch Λ (for example, =2.3 µm by default) as shown in Figure 4 (for a 11-by-5 circle PCF). The waveguide project was built using the half structure and then taking advantage of the geometric symmetry. The whole structure, however, is required to simulate the performance parameters related to the waveguide bend. 4.3 Solver settings for the waveguide After the creation of the PCF, the following step is to select the appropriate solver setting for simulation by clicking button. Figure 6 shows the Waveguide Solver Setting window, which consists of two tabs: General Information and FD Mode Solver Setting. Under the General Information tab, the user can: select appropriate methods, such as Simulation or Scan choose suitable wavelength range and points select the required Mesh setting, as shown in Figure 6 where a Bi-Linear method is used. Apollo Inc. Page 17 of 21 APN-APSS-PCF

18 In order to achieve accurate results, make sure that the mesh boundaries are both coincident with the dielectric boundaries (for example, use the multi-section mesh button), as well as being similar in size in both directions. If Scan is selected, Variable Selection must also be used to specify appropriate settings for the scan. Users can also perform a Structure Check for selecting scan parameters. In the current version, the maximum number of the variable scan is two. The FD Mode Solver Setting tab has two sub-tabs: General Setting and Advanced Setting. The General Setting tab, as shown in Figure 7, allows the user to select the appropriate settings depending on accuracy requirements and time constraints. Note: For more detailed information about the types of solvers and settings, refer to APSS User Manual. Figure 6 The waveguide solver setting and mesh setting editor Apollo Inc. Page 18 of 21 APN-APSS-PCF

19 Figure 7 The FD mode solver setting and PML number of layer 4.4 Run and Display After selecting the solver settings, the user can simulate the PCF by clicking the Run button. After finishing the simulation, the user can view the S parameters and EM fields at the positions defined in the Section Position by clicking the or button. The calculated dispersion curves for both X- and Y-polarizations and the modal profile E x for the X-polarization are shown in Figure 8 and Figure 9, respectively. They agree well with experimental and simulated results achieved by other numerical methods [5]. For some modal properties, such as the dispersion, which is highly sensitive to the accuracy of the modal calculation (for example, the second derivative of the real part of the effective index), the simulation accuracy depends on having the required solver setting. Especially for the small d/ at long wavelengths, the larger number of air rings and dense meshes must be selected. Apollo Inc. Page 19 of 21 APN-APSS-PCF

20 Figure 10 shows the waveguide dispersion D g (without considering the material dispersion and set index of silica to equal 1.45) as a function of wavelength for PCFs with different d/λ values with fixed Λ=2.3 µm. Figure 8 The effective index of the PCF with d =1.84 µm and Λ=2.3 µm (a) d=0.46 µm (b) d = 0.92 µm (c) d = 1.84 µm Apollo Inc. Page 20 of 22 APN-APSS-PCF

21 Figure 9 The modal profile E x of the PCF with the pitch Λ=2.3 µm at wavelength 1.55 µm Waveguide dispersion D g (ps/nm/km) (b) d=0.460µm d=0.805µm d=1.000µm d=1.380µm -350 d=1.840µm -400 d=2.300µm Wavelength λ (µm) Figure 10 Waveguide dispersion D g as a function of wavelength for PCFs with different d/λ values with fixed Λ=2.3 µm By using flexible simulation and scan function, it is convenient to do the sensitivity analysis of the PCF device related design parameters such as air hole size and pitch, as well as, polarization, and wavelength dependence. 5 Conclusion In using practical examples from the references, we have shown that it is possible to design and simulate the PCF by taking advantage of the user-friendly user-defined model in APSS Waveguide Module. The theory and operational principle of the PCF have been described. Finally, the design procedure, starting from material to waveguide has been illustrated, by taking some practical examples from the references. The simulation results agree well with experimental results. Apollo Inc. Page 21 of 22 APN-APSS-PCF

22 6 References [1] J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, Photonic Crystal Fibers: A New Class of Optical Waveguides, Opt. Fib. Technol., vol. 5, pp , [2] J. D. Joannopoulos, Photonics Crystals: Molding the flow of the light, New Jersey: Princeton University Press, [3] L. P. Shen, W.P. Huang, and S. S. Jian, Design of photonic crystal fibers for dispersion-related applications, J. Lightwave Technol., vol. 21, no.7, [4] G. Agrawal, Nonlinear Fiber Optics. New York: Academic, [5] J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, All-silica single-mode optical fiber with photonic crystal cladding, Opt. Lett., vol. 21, pp , 1996; Also, errata, vol. 22, pp , Apollo Inc. Page 22 of 22 APN-APSS-PCF